TPTP Problem File: NUN014_5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : NUN014_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 197
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_197 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 171 ( 46 unt; 39 typ; 0 def)
% Number of atoms : 308 ( 64 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 215 ( 39 ~; 6 |; 6 &)
% ( 31 <=>; 133 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 13 ( 9 >; 4 *; 0 +; 0 <<)
% Number of predicates : 22 ( 21 usr; 0 prp; 1-3 aty)
% Number of functors : 15 ( 15 usr; 6 con; 0-4 aty)
% Number of variables : 258 ( 225 !; 4 ?; 258 :)
% ( 29 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:26:31
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (35)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Osgn__if,type,
sgn_if:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__vector,type,
real_normed_vector:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim1727834104eiling:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__algebra__1,type,
real_n2089651433ebra_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
tff(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim2085082626_floor:
!>[A: $tType] : ( A > int ) ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B1: $tType] : ( ( fun(A,B1) * A ) > B1 ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_m1____,type,
m1: int ).
tff(sy_v_n____,type,
n: nat ).
tff(sy_v_t____,type,
t: int ).
tff(sy_v_tn____,type,
tn: nat ).
%----Relevant facts (98)
tff(fact_0_zero__less__ceiling,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Xa: A] :
( ord_less(int,zero_zero(int),archimedean_ceiling(A,Xa))
<=> ord_less(A,zero_zero(A),Xa) ) ) ).
tff(fact_1_floor__less__zero,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Xa: A] :
( ord_less(int,archim2085082626_floor(A,Xa),zero_zero(int))
<=> ord_less(A,Xa,zero_zero(A)) ) ) ).
tff(fact_2_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ~ ord_less(A,X,X) ) ).
tff(fact_3_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Ya: A,Xa: A] :
( ( Xa != Ya )
<=> ( ord_less(A,Xa,Ya)
| ord_less(A,Ya,Xa) ) ) ) ).
tff(fact_4_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Ya: A,Xa: A] :
( ~ ord_less(A,Xa,Ya)
<=> ( ord_less(A,Ya,Xa)
| ( Xa = Ya ) ) ) ) ).
tff(fact_5_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
| ( X = Y )
| ord_less(A,Y,X) ) ) ).
tff(fact_6_linorder__antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xa: A,Ya: A] :
( ~ ord_less(A,Ya,Xa)
=> ( ~ ord_less(A,Xa,Ya)
<=> ( Xa = Ya ) ) ) ) ).
tff(fact_7_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ( X != Y )
=> ( ~ ord_less(A,X,Y)
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_8_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X: A] :
( ( X != Y )
=> ( ~ ord_less(A,X,Y)
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_9_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( X != Y ) ) ) ).
tff(fact_10_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ~ ord_less(A,Y,X) ) ) ).
tff(fact_11_n0,axiom,
ord_less(nat,zero_zero(nat),n) ).
tff(fact_12_floor__zero,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ( archim2085082626_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).
tff(fact_13_ceiling__zero,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).
tff(fact_14_floor__less__cancel,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Y: A,X: A] :
( ord_less(int,archim2085082626_floor(A,X),archim2085082626_floor(A,Y))
=> ord_less(A,X,Y) ) ) ).
tff(fact_15_ceiling__less__cancel,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Y: A,X: A] :
( ord_less(int,archimedean_ceiling(A,X),archimedean_ceiling(A,Y))
=> ord_less(A,X,Y) ) ) ).
tff(fact_16_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [Xa: A] :
( ( zero_zero(A) = Xa )
<=> ( Xa = zero_zero(A) ) ) ) ).
tff(fact_17_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ ord_less(A,X,Y)
=> ( ( X != Y )
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_18_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ~ ord_less(A,Y,X) ) ) ).
tff(fact_19_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X: A] :
( ord_less(A,X,Y)
=> ( ord_less(A,Y,Z)
=> ord_less(A,X,Z) ) ) ) ).
tff(fact_20_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B2: A,A2: A] :
( ord_less(A,A2,B2)
=> ( ( B2 = C1 )
=> ord_less(A,A2,C1) ) ) ) ).
tff(fact_21_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B2: A,A2: A] :
( ( A2 = B2 )
=> ( ord_less(A,B2,C1)
=> ord_less(A,A2,C1) ) ) ) ).
tff(fact_22_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A2: A] :
( ord_less(A,A2,B2)
=> ~ ord_less(A,B2,A2) ) ) ).
tff(fact_23_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [P: bool,Ya: A,Xa: A] :
( ord_less(A,Xa,Ya)
=> ( ord_less(A,Ya,Xa)
=> pp(P) ) ) ) ).
tff(fact_24_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( Y != X ) ) ) ).
tff(fact_25_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( X != Y ) ) ) ).
tff(fact_26_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ~ ord_less(A,Y,X) ) ) ).
tff(fact_27_order__less__subst1,axiom,
! [A: $tType,B1: $tType] :
( ( order(B1)
& order(A) )
=> ! [C: B1,B: B1,F: fun(B1,A),A1: A] :
( ord_less(A,A1,aa(B1,A,F,B))
=> ( ord_less(B1,B,C)
=> ( ! [X2: B1,Y1: B1] :
( ord_less(B1,X2,Y1)
=> ord_less(A,aa(B1,A,F,X2),aa(B1,A,F,Y1)) )
=> ord_less(A,A1,aa(B1,A,F,C)) ) ) ) ) ).
tff(fact_28_ord__eq__less__subst,axiom,
! [A: $tType,B1: $tType] :
( ( ord(B1)
& ord(A) )
=> ! [C: B1,B: B1,F: fun(B1,A),A1: A] :
( ( A1 = aa(B1,A,F,B) )
=> ( ord_less(B1,B,C)
=> ( ! [X2: B1,Y1: B1] :
( ord_less(B1,X2,Y1)
=> ord_less(A,aa(B1,A,F,X2),aa(B1,A,F,Y1)) )
=> ord_less(A,A1,aa(B1,A,F,C)) ) ) ) ) ).
tff(fact_29_order__less__subst2,axiom,
! [A: $tType,B1: $tType] :
( ( order(B1)
& order(A) )
=> ! [C: B1,F: fun(A,B1),B: A,A1: A] :
( ord_less(A,A1,B)
=> ( ord_less(B1,aa(A,B1,F,B),C)
=> ( ! [X2: A,Y1: A] :
( ord_less(A,X2,Y1)
=> ord_less(B1,aa(A,B1,F,X2),aa(A,B1,F,Y1)) )
=> ord_less(B1,aa(A,B1,F,A1),C) ) ) ) ) ).
tff(fact_30_ord__less__eq__subst,axiom,
! [A: $tType,B1: $tType] :
( ( ord(B1)
& ord(A) )
=> ! [C: B1,F: fun(A,B1),B: A,A1: A] :
( ord_less(A,A1,B)
=> ( ( aa(A,B1,F,B) = C )
=> ( ! [X2: A,Y1: A] :
( ord_less(A,X2,Y1)
=> ord_less(B1,aa(A,B1,F,X2),aa(A,B1,F,Y1)) )
=> ord_less(B1,aa(A,B1,F,A1),C) ) ) ) ) ).
tff(fact_31_dense,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ? [Z2: A] :
( ord_less(A,X,Z2)
& ord_less(A,Z2,Y) ) ) ) ).
tff(fact_32_lt__ex,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X: A] :
? [Y1: A] : ord_less(A,Y1,X) ) ).
tff(fact_33_gt__ex,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Xa: A] :
? [X1: A] : ord_less(A,Xa,X1) ) ).
tff(fact_34_sgn__less,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ord_less(A,sgn_sgn(A,A1),zero_zero(A))
<=> ord_less(A,A1,zero_zero(A)) ) ) ).
tff(fact_35_sgn__greater,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ord_less(A,zero_zero(A),sgn_sgn(A,A1))
<=> ord_less(A,zero_zero(A),A1) ) ) ).
tff(fact_36_tn0,axiom,
ord_less(nat,zero_zero(nat),tn) ).
tff(fact_37_sgn__sgn,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] : sgn_sgn(A,sgn_sgn(A,A2)) = sgn_sgn(A,A2) ) ).
tff(fact_38_sgn0,axiom,
! [A: $tType] :
( sgn_if(A)
=> ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_39_sgn__0__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ( sgn_sgn(A,A1) = zero_zero(A) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_40_sgn__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_41_neq0__conv,axiom,
! [Na: nat] :
( ( Na != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),Na) ) ).
tff(fact_42_less__nat__zero__code,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_43_less__zeroE,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_44_sgn__zero__iff,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [Xa: A] :
( ( sgn_sgn(A,Xa) = zero_zero(A) )
<=> ( Xa = zero_zero(A) ) ) ) ).
tff(fact_45_nat__less__cases,axiom,
! [P: fun(nat,fun(nat,bool)),Na: nat,Ma: nat] :
( ( ord_less(nat,Ma,Na)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,Na),Ma)) )
=> ( ( ( Ma = Na )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,Na),Ma)) )
=> ( ( ord_less(nat,Na,Ma)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,Na),Ma)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,Na),Ma)) ) ) ) ).
tff(fact_46_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_47_less__not__refl2,axiom,
! [M: nat,N: nat] :
( ord_less(nat,N,M)
=> ( M != N ) ) ).
tff(fact_48_less__irrefl__nat,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_49_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ ord_less(nat,X,Y)
=> ord_less(nat,Y,X) ) ) ).
tff(fact_50_nat__neq__iff,axiom,
! [Na: nat,Ma: nat] :
( ( Ma != Na )
<=> ( ord_less(nat,Ma,Na)
| ord_less(nat,Na,Ma) ) ) ).
tff(fact_51_less__not__refl,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_52_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_53_gr__implies__not0,axiom,
! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ( N != zero_zero(nat) ) ) ).
tff(fact_54_not__less0,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_55_of__nat__0__less__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Na: nat] :
( ord_less(A,zero_zero(A),semiring_1_of_nat(A,Na))
<=> ord_less(nat,zero_zero(nat),Na) ) ) ).
tff(fact_56_infinite__descent0,axiom,
! [Na: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,P,zero_zero(nat)))
=> ( ! [N1: nat] :
( ord_less(nat,zero_zero(nat),N1)
=> ( ~ pp(aa(nat,bool,P,N1))
=> ? [M1: nat] :
( ord_less(nat,M1,N1)
& ~ pp(aa(nat,bool,P,M1)) ) ) )
=> pp(aa(nat,bool,P,Na)) ) ) ).
tff(fact_57_sgn__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] :
( ord_less(A,zero_zero(A),A2)
=> ( sgn_sgn(A,A2) = one_one(A) ) ) ) ).
tff(fact_58_sgn__1__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ( sgn_sgn(A,A1) = one_one(A) )
<=> ord_less(A,zero_zero(A),A1) ) ) ).
tff(fact_59_t1,axiom,
ord_less(int,one_one(int),t) ).
tff(fact_60_of__nat__eq__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [Na: nat,Ma: nat] :
( ( semiring_1_of_nat(A,Ma) = semiring_1_of_nat(A,Na) )
<=> ( Ma = Na ) ) ) ).
tff(fact_61_of__nat__1,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,one_one(nat)) = one_one(A) ) ) ).
tff(fact_62_floor__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ( archim2085082626_floor(A,one_one(A)) = one_one(int) ) ) ).
tff(fact_63_ceiling__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).
tff(fact_64_sgn__one,axiom,
! [A: $tType] :
( real_n2089651433ebra_1(A)
=> ( sgn_sgn(A,one_one(A)) = one_one(A) ) ) ).
tff(fact_65_floor__of__nat,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [N: nat] : archim2085082626_floor(A,semiring_1_of_nat(A,N)) = semiring_1_of_nat(int,N) ) ).
tff(fact_66_ceiling__of__nat,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [N: nat] : archimedean_ceiling(A,semiring_1_of_nat(A,N)) = semiring_1_of_nat(int,N) ) ).
tff(fact_67_of__nat__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,zero_zero(nat)) = zero_zero(A) ) ) ).
tff(fact_68_of__nat__less__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Na: nat,Ma: nat] :
( ord_less(A,semiring_1_of_nat(A,Ma),semiring_1_of_nat(A,Na))
<=> ord_less(nat,Ma,Na) ) ) ).
tff(fact_69_floor__less__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Xa: A] :
( ord_less(int,archim2085082626_floor(A,Xa),one_one(int))
<=> ord_less(A,Xa,one_one(A)) ) ) ).
tff(fact_70_one__less__ceiling,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Xa: A] :
( ord_less(int,one_one(int),archimedean_ceiling(A,Xa))
<=> ord_less(A,one_one(A),Xa) ) ) ).
tff(fact_71_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
tff(fact_72_one__neq__zero,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( one_one(A) != zero_zero(A) ) ) ).
tff(fact_73_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [Xa: A] :
( ( one_one(A) = Xa )
<=> ( Xa = one_one(A) ) ) ) ).
tff(fact_74_of__nat__less__0__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M: nat] : ~ ord_less(A,semiring_1_of_nat(A,M),zero_zero(A)) ) ).
tff(fact_75_of__nat__less__imp__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,M: nat] :
( ord_less(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N))
=> ord_less(nat,M,N) ) ) ).
tff(fact_76_less__imp__of__nat__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ord_less(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N)) ) ) ).
tff(fact_77_not__one__less__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ~ ord_less(A,one_one(A),zero_zero(A)) ) ).
tff(fact_78_zero__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ord_less(A,zero_zero(A),one_one(A)) ) ).
tff(fact_79_mn,axiom,
ord_less(int,m1,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))) ).
tff(fact_80_zero__less__int__conv,axiom,
! [Na: nat] :
( ord_less(int,zero_zero(int),semiring_1_of_nat(int,Na))
<=> ord_less(nat,zero_zero(nat),Na) ) ).
tff(fact_81_n1pos,axiom,
ord_less(int,zero_zero(int),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))) ).
tff(fact_82_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,A1: A,B: A] :
( ( plus_plus(A,B,A1) = plus_plus(A,C,A1) )
<=> ( B = C ) ) ) ).
tff(fact_83_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,B: A,A1: A] :
( ( plus_plus(A,A1,B) = plus_plus(A,A1,C) )
<=> ( B = C ) ) ) ).
tff(fact_84_double__eq__0__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ( plus_plus(A,A1,A1) = zero_zero(A) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_85_double__zero__sym,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ( zero_zero(A) = plus_plus(A,A1,A1) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_86_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,C: A,A1: A] :
( ord_less(A,plus_plus(A,A1,C),plus_plus(A,B,C))
<=> ord_less(A,A1,B) ) ) ).
tff(fact_87_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,A1: A,C: A] :
( ord_less(A,plus_plus(A,C,A1),plus_plus(A,C,B))
<=> ord_less(A,A1,B) ) ) ).
tff(fact_88_of__nat__add,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat,M: nat] : semiring_1_of_nat(A,plus_plus(nat,M,N)) = plus_plus(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N)) ) ).
tff(fact_89_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ord_less(A,plus_plus(A,A1,A1),zero_zero(A))
<=> ord_less(A,A1,zero_zero(A)) ) ) ).
tff(fact_90_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ord_less(A,zero_zero(A),plus_plus(A,A1,A1))
<=> ord_less(A,zero_zero(A),A1) ) ) ).
tff(fact_91_floor__add__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X: A] : archim2085082626_floor(A,plus_plus(A,X,one_one(A))) = plus_plus(int,archim2085082626_floor(A,X),one_one(int)) ) ).
tff(fact_92_ceiling__add__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X: A] : archimedean_ceiling(A,plus_plus(A,X,one_one(A))) = plus_plus(int,archimedean_ceiling(A,X),one_one(int)) ) ).
tff(fact_93_int__int__eq,axiom,
! [Na: nat,Ma: nat] :
( ( semiring_1_of_nat(int,Ma) = semiring_1_of_nat(int,Na) )
<=> ( Ma = Na ) ) ).
tff(fact_94_int__1,axiom,
semiring_1_of_nat(int,one_one(nat)) = one_one(int) ).
tff(fact_95_odd__less__0__iff,axiom,
! [Z1: int] :
( ord_less(int,plus_plus(int,plus_plus(int,one_one(int),Z1),Z1),zero_zero(int))
<=> ord_less(int,Z1,zero_zero(int)) ) ).
tff(fact_96_zless__add1__eq,axiom,
! [Z1: int,Wa: int] :
( ord_less(int,Wa,plus_plus(int,Z1,one_one(int)))
<=> ( ord_less(int,Wa,Z1)
| ( Wa = Z1 ) ) ) ).
tff(fact_97_odd__nonzero,axiom,
! [Z: int] : plus_plus(int,plus_plus(int,one_one(int),Z),Z) != zero_zero(int) ).
%----Arities (31)
tff(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(int) ).
tff(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
tff(arity_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(int) ).
tff(arity_Int_Oint___Orderings_Opreorder,axiom,
preorder(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Orderings_Oorder,axiom,
order(int) ).
tff(arity_Int_Oint___Orderings_Oord,axiom,
ord(int) ).
tff(arity_Int_Oint___Groups_Osgn__if,axiom,
sgn_if(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Groups_Oone,axiom,
one(int) ).
tff(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
tff(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
tff(arity_Nat_Onat___Nat_Osemiring__char__0,axiom,
semiring_char_0(nat) ).
tff(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one(nat) ).
tff(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
tff(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
tff(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (1)
tff(conj_0,conjecture,
ord_less(int,zero_zero(int),m1) ).
%------------------------------------------------------------------------------